# 演習問題

$z=f\left(x,y\right)\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=t-sint\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=1-cost\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ のとき， $\text{\hspace{0.17em}}\frac{{d}^{2}z}{d{t}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ を求めよ．

2次の偏微分

$z=f\left(x,y\right)$$x=2{t}^{2}-3$$y={t}^{2}+3t+7$ のとき， $\frac{{d}^{2}z}{d{t}^{2}}$ を求めよ．

$z=f\left(x,y\right),x=u\mathrm{cos}\theta -v\mathrm{sin}\theta ,$ $y=u\mathrm{sin}\theta +v\mathrm{cos}\theta$ のとき

$\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=\frac{{\partial }^{2}z}{\partial {u}^{2}}+\frac{{\partial }^{2}z}{\partial {v}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

となることを示せ．

$z=f\left(x,y\right)\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=r\mathrm{cos}\theta \text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=r\mathrm{sin}\theta \text{\hspace{0.17em}}\text{\hspace{0.17em}}$ のとき

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=\frac{{\partial }^{2}z}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial z}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}z}{\partial {\theta }^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

となることを示せ．

$z$ に関する方程式

$\frac{{\partial }^{2}z}{\partial {t}^{2}}={c}^{2}\left(\frac{{\partial }^{2}z}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial z}{\partial r}\right)$

において， $z=\frac{1}{r}u$ とおき， $u$ に関する方程式に変換すると

$\frac{{\partial }^{2}u}{\partial {t}^{2}}={c}^{2}\frac{{\partial }^{2}u}{\partial {r}^{2}}$

となることを示せ．